Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


282       XIV    INTEGRATION IN  LOCALLY COMPACT GROUPS

In particular, for each x e G and each locally /?-integrable function /,
the convolutions zx */and/* sx are defined and (with the above conventions)
are given by the formulas

(14.8.5)       (e, */)00 =

9. EXAMPLES OF CONVOLUTIONS OF MEASURES AND FUNCTIONS

Generally speaking, if we are looking for usable sufficient conditions for a
measure /z and a function/on G to be convolvable, the stronger the conditions
imposed on one of the factors, the weaker the conditions that need to be imposed
on the other. Moreover, the function \JL */is "at least as regular" as/, in
general. (For a Lie group G, we shall obtain these characteristics of con-
volution in a more general context in Chapter XVII.)

(14.9.1) A measure ILL with compact support is convolvable with every locally
fi-integrable function f. If in addition f is continuous (resp. continuous with
compact support), then the integral on the right-hand side of (14.8.2) is defined
for all x e G, and the function

.-{A.-.

is continuous (resp. continuous with compact support).

The first assertion is a particular case of (14.6.4), and the fact that
s*-+f(s~'1x) is ^-integrable is a consequence of (13.19.3). The continuity of
/z*/when/is continuous follows from (14.1.5.5). Finally, that /**/has
compact support when/has compact support is a particular case of (14.5.4).

(14.9.2)   Let u be a bounded measure on G.

(i) For p = 1, 2 or + oo, the measure \JL is convolvable with every function
/J5?(G, /?). The function u * f (defined almost everywhere by (14.8.2)) then
belongs to ^(G, /?), and we have

(ii) Iff is continuous and bounded, then the integral on the right-hand
side of (14.8.2) is defined for all x e G, and the function x^(f(s"lx) dfi(s) is
continuous and bounded on G.

(iii)   Iffe 2(G) (13.20.5), then p */e <fg(G).hat